People enrolled in new plans under the health law are showing higher rates of serious health conditions than other insurance customers, according to an early analysis of medical claims, putting pressure on insurers around the country as they prepare to propose rates for next year.

Among those health-law marketplace enrollees who have seen a doctor or other health-care provider in the first quarter of this year, around 27% have significant health issues such as diabetes, psychiatric conditions, asthma, heart problems or...